

where \(L_{\mathrm {int}} =4\pi r^2 {\sigma }{T_{\textnormal {int}}}^4\)  is the interior luminosity from the planet.
[1]} showed that a constant \(L(r)\)  is a good approximation. However, Equation (REF ) does not reflect the region of a flatter, almost constant, \(T(r)\)  profile where the dust is destroyed, near a destruction temperature \({T_{\textnormal {dest}}}=1220\times {\rho _{-11}}^{0.0195}\)  K, where \(\rho _{-11}\equiv \rho \times 10^{11}~\mathrm {cm}^3\,\mathrm {g}^{-1}\)  [2]}. In particular, Equation (REF ) assumes that the frequency-averaged radiation is free-streaming throughout. Thus, if the accretion flow is devoid of dust, Equation (REF ) will hold, while in the presence of dust this will slightly underestimate the temperature at a given radial position (and thus density). However, at these low temperatures \(T\lesssim 1200\)  K, the gas opacity is small anyway and we find (see Section ) that, in the relevant part of the parameter space, the H \(\alpha \)  is not extincted. This justifies approximately the simplification.
